This work aims to construct an efficient and highly accurate numerical method to address the time singularity at $t=0$ involved in a class of time-fractional parabolic integro-partial differential equations in one and two dimensions. The $L2$-$1_\sigma$ scheme is used to discretize the time-fractional operator, whereas a modified version of the composite trapezoidal approximation is employed to discretize the Volterra operator. Subsequently, it helps to convert the proposed model into a second-order BVP in a semi-discrete form. The multi-dimensional Haar wavelets are then used for grid adaptation and efficient computations for the 2D problem, whereas the standard second-order approximations are employed to approximate the spatial derivatives for the 1D case. The stability analysis is carried out on an adaptive mesh in time. The convergence analysis leads to $O(N^{-2}+M^{-2})$ accurate solution in the space-time domain for the 1D problem having time singularity based on the $L^\infty$ norm for a suitable choice of the grading parameter. Furthermore, it provides $O(N^{-2}+M^{-3})$ accurate solution for the 2D problem having unbounded time derivative at $t=0$. The analysis also highlights a higher order accuracy for a sufficiently smooth solution resides in $C^3(\overline{\Omega}_t)$ even if the mesh is discretized uniformly. The truncation error estimates for the time-fractional operator, integral operator, and spatial derivatives are presented. Numerous tests are performed on several examples in support of the theoretical analysis. The advancement of the proposed methodology is demonstrated through the application of the time-fractional Fokker-Planck equation and the fractional-order viscoelastic dynamics having weakly singular kernels. It also confirms the superiority of the proposed method compared with existing approaches available in the literature.

翻译：本研究旨在构建一种高效且高精度的数值方法，以处理一维和二维时间分数阶抛物型积分-偏微分方程在$t=0$处存在的时间奇异性。采用$L2$-$1_\sigma$格式离散时间分数阶算子，而采用修正的复合梯形近似法离散Volterra算子。随后，这有助于将所提模型转化为半离散形式的二阶边值问题。对于二维问题，采用多维Haar小波进行网格自适应和高效计算；对于一维情况，则采用标准二阶近似法逼近空间导数。在自适应时间网格上进行了稳定性分析。收敛性分析表明：基于$L^\infty$范数，对于具有时间奇异性的二维问题，在适当选择分级参数时，时空域内可获得$O(N^{-2}+M^{-2})$精度的解；对于在$t=0$处具有无界时间导数的二维问题，则可获得$O(N^{-2}+M^{-3})$精度的解。分析还指出，即使采用均匀网格离散，对于存在于$C^3(\overline{\Omega}_t)$中的充分光滑解，该方法仍能保持更高阶精度。文中给出了时间分数阶算子、积分算子及空间导数的截断误差估计。通过多个算例进行了大量测试以验证理论分析。将所提方法应用于时间分数阶Fokker-Planck方程和具有弱奇异核的分数阶粘弹性动力学问题，展示了该方法的先进性。与文献中现有方法相比，进一步证实了所提方法的优越性。